(see [1]}).
Therefore, \(\zeta \)  is well-defined (and, as it turns out, continuous)
as a map \(\zeta \colon {A}^n\rightarrow (\mathbb {R}/2\pi \mathbb {Z})^{2\mathfrak {g}}\) .
If for some \(j\)  and \(k\)  the curve \(\gamma _k\)  contains \(a_j\) , then we consider
a small perturbation \(\lambda _k\)  of \(\gamma _k\) , which is still
a smooth, closed, simple curve, is homologous to \(\gamma _k\) 
but avoids all the points \(a_j\) . We define \(\zeta _k(\mathbf {a}, \, \mathbf {d})\) 
by integrating over \(\lambda _k\)  instead of \(\gamma _k\) .
The set \(\mathcal {L}(\mathbf {a}, \, \mathbf {d})\)  in Theorem REF 
is defined in terms of \(\alpha \)  and \(\mathcal {L}(\mathbf {a}, \, \mathbf {d})\) ,
as follows:
\( \mathcal {L}(\mathbf {a}, \, \mathbf {d}) := \left\lbrace \xi \in \mathrm {Harm}^1(M)\colon \alpha (\xi )+ \zeta (\mathbf {a}, \, \mathbf {d}) \in (2\pi \mathbb {Z})^{2\mathfrak {g}} \right\rbrace \) 
